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Discrete Integrable Systems: QRT Maps and Elliptic Surfaces

TL;DR: The QRT Map as discussed by the authors is the pencil of biquadratic curves in the projective plane of the QRT surface and is used to measure the distance between two points.
Abstract: The QRT Map.- The Pencil of Biquadratic Curves in .- The QRT surface.- Cubic Curves in the Projective Plane.- The Action of the QRT Map on Homology.- Elliptic Surfaces.- Automorphisms of Elliptic Surfaces.- Elliptic Fibrations with a Real Structure.- Rational elliptic surfaces.- Symmetric QRT Maps.- Examples from the Literature.- Appendices.
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TL;DR: The Tate-Shafarevich group of genus-one fibrations as discussed by the authors has been shown to have an F-theory limit when the area of the genus one fiber approaches zero.
Abstract: We argue that M-theory compactified on an arbitrary genus-one fibration, that is, an elliptic fibration which need not have a section, always has an F-theory limit when the area of the genus-one fiber approaches zero. Such genus-one fibrations can be easily constructed as toric hypersurfaces, and various SU(5) × U(1) n and E 6 models are presented as examples. To each genus-one fibration one can associate a τ -function on the base as well as an SL(2, $$ \mathbb{Z} $$ ) representation which together define the IIB axio-dilaton and 7-brane content of the theory. The set of genus-one fibrations with the same τ -function and SL(2, $$ \mathbb{Z} $$ ) representation, known as the Tate-Shafarevich group, supplies an important degree of freedom in the corresponding F-theory model which has not been studied carefully until now. Six-dimensional anomaly cancellation as well as Witten’s zero-mode count on wrapped branes both imply corrections to the usual F-theory dictionary for some of these models. In particular, neutral hypermultiplets which are localized at codimension-two fibers can arise. (All previous known examples of localized hypermultiplets were charged under the gauge group of the theory.) Finally, in the absence of a section some novel monodromies of Kodaira fibers are allowed which lead to new breaking patterns of non-Abelian gauge groups.

175 citations

Journal ArticleDOI
TL;DR: In this paper, the dynamics of systems describing the rolling without slipping and spinning (rubber rolling) of various rigid bodies on a plane and a sphere are investigated, and a hierarchy of possible types of dynamical behavior arises depending on the body's surface geometry and mass distribution.
Abstract: In this paper, we investigate the dynamics of systems describing the rolling without slipping and spinning (rubber rolling) of various rigid bodies on a plane and a sphere. It is shown that a hierarchy of possible types of dynamical behavior arises depending on the body’s surface geometry and mass distribution. New integrable cases and cases of existence of an invariant measure are found. In addition, these systems are used to illustrate that the existence of several nontrivial involutions in reversible dissipative systems leads to quasi-Hamiltonian behavior.

112 citations

Journal ArticleDOI
TL;DR: In this paper, the authors present a general algorithm for computing the Weierstrass form of elliptic curves defined as complete intersections of different codimensions and use it to solve all cases of complete intersections in an ambient toric variety, using this result, they determine the toric Mordell-Weil groups of all 3134 nef partitions obtained from the 4319 threedimensional reflexive polytopes and find new groups that do not exist for toric hypersurfaces.
Abstract: Global F-theory compactifications whose fibers are realized as complete inter-sections form a richer set of models than just hypersurfaces. The detailed study of the physics associated with such geometries depends crucially on being able to put the elliptic fiber into Weierstrass form. While such a transformation is always guaranteed to exist, its explicit form is only known in a few special cases. We present a general algorithm for computing the Weierstrass form of elliptic curves defined as complete intersections of different codimensions and use it to solve all cases of complete intersections of two equations in an ambient toric variety. Using this result, we determine the toric Mordell-Weil groups of all 3134 nef partitions obtained from the 4319 three-dimensional reflexive polytopes and find new groups that do not exist for toric hypersurfaces. As an application, we construct several models that cannot be realized as toric hypersurfaces, such as the first toric SU(5) GUT model in the literature with distinctly charged 10 representations and an F-theory model with discrete gauge group ℤ4 whose dual fiber has a Mordell-Weil group with ℤ4 torsion.

99 citations

Posted Content
29 Sep 2015
TL;DR: In this article, a comprehensive review is given on the current status of achievements in the geometric aspects of the Painlev\'e equations, with a particular emphasis on the discrete Painlev'e equations.
Abstract: In this paper a comprehensive review is given on the current status of achievements in the geometric aspects of the Painlev\'e equations, with a particular emphasis on the discrete Painlev\'e equations. The theory is controlled by the geometry of certain rational surfaces called the spaces of initial values, which are characterized by eight point configuration on $\mathbb{P}^1\times\mathbb{P}^1$ and classified according to the degeration of points. We give a systematic description of the equations and their various properties, such as affine Weyl group symmetries, hypergeomtric solutions and Lax pairs under this framework, by using the language of Picard lattice and root systems. We also provide with a collection of basic data; equations, point configurations/root data, Weyl group representations, Lax pairs, and hypergeometric solutions of all possible cases.

77 citations

Journal ArticleDOI
TL;DR: In this paper, it was shown that Kahan's discretization of quadratic vector fields is equivalent to a Runge-Kutta method, which produces large classes of integrable rational mappings in two and three dimensions.
Abstract: We show that Kahan's discretization of quadratic vector fields is equivalent to a Runge–Kutta method. When the vector field is Hamiltonian on either a symplectic vector space or a Poisson vector space with constant Poisson structure, the map determined by this discretization has a conserved modified Hamiltonian and an invariant measure, a combination previously unknown amongst Runge–Kutta methods applied to nonlinear vector fields. This produces large classes of integrable rational mappings in two and three dimensions, explaining some of the integrable cases that were previously known.

71 citations